IntMath Newsletter: radius of curvature, log curve, free math videos

By Murray Bourne, 23 Jul 2010

22 Jul 2010

In this Newsletter:

1. Math of the great summer brain drain
2. Math tip: Radius of Curvature, an application of differentiation
3. Slope of the logarithm curve at any point
4. Friday math movie - Khan's Academy
5. How to import GeoGebra files into JSXGraph
6. Final thought – stay positive

1. Math of the great summer brain drain

Here is some interesting information about how forgetting works, and what we can do to reduce it.

Do you want to be a successful math student? This could be a good place to start.

Suitable for: Everyone.


Everyone knows students drop a few grades in their knowledge during the summer months. What's the math behind this?

Read more: Math of the great summer brain drain

2. Math tip: Radius of Curvature, an application of differentiation

I re-wrote a page on Interactive Mathematics recently and thought you may find it useful.

A reader sent in a question asking how to find the radius of curvature if we don't know the function. I solve this problem in 3 different ways. You can get some good mathematical understanding from this.

Suitable for: Everyone. Even if you have never heard of calculus yet, you will come across it one day and this example gives you an idea about what it can be used for.

Summary of the page:

  • What is radius of curvature and why study it?
  • Example of radius of curvature involving a cubic curve
  • Exploration of radius of curvature using an interactive graph
  • A second example (the one sent in by a reader) that is solved in 3 different ways:
    1. Using a parabola to model the data and differentiation
    2. Using linear approximations and concepts of calculus
    3. Using the formula for the circle passing through 3 points

So here it is:

train tracks radius of curvature

In this example, we learn about radius of curvature and one of its applications.

Read more: Radius of Curvature

3. Slope of the logarithm curve at any point

Suitable for: Everyone. Once again, this example involves differentiation (a calculus topic), but the main concept is easy for everyone to understand: as you move around a curve, the slope changes.

Geogebra to JSXGraph

This is a new interactive graph in IntMath. You can drag a point around a logarithm curve to learn about the changing slope.

You'll laso learn what a log curve looks like.

Learn more: Slope of the Logarithm Function

4. Friday math movie - Khan's Academy

Suitable for: Everyone.

Khan Academy

Here's a wide range of free videos by a guy that clearly loves to teach. This is a very popular math video series.

Watch a typical video. This one is about the Unit Circle:

Friday math movie – free math videos by Khan Academy

5. How to import GeoGebra files into JSXGraph

Suitable for: Math computer geeks.

JSXGraph and Geogebra

This article explains how you can create interactive math applets using GeoGebra (which is easy) and import those files into JSXGraph (which displays the math in a browser, and is cross-platform and cross-browser).

Read more: How to import GeoGebra files into JSXGraph

6. Final thought – stay positive

A lot has been written about the power of positive thinking. But it's true!

The people who have a negative approach to math ("This is dumb!" "This is never used in the real world!") and do nothing much to improve their skills, get into a cycle where they get worse.

Here's a thought from self-help speaker and author Brian Tracy:

"The more positive you are when you think and work toward your goals, the faster you achieve them." [Brian Tracy]

Until next time, enjoy whatever you learn.

See the 16 Comments below.

16 Comments on “IntMath Newsletter: radius of curvature, log curve, free math videos”

  1. Geoff says:


    Thanks so much for your web site - as a teacher of maths (in engineering contaxt) I can say that your approach fits well with me and no doubt my students.

    I thought you might be interested in this, on positive thinking:

    Geoff, Perth, Australia

  2. Murray says:

    Thanks for the interesting link, Geoff.

    One of the downsides of optimism is the related issue of positive self image. If we regard positive self image as being more important than actual achievement, it becomes counter-productive and de-motivational.

    It's all about balance, as usual.

  3. Elena Nava-Garcia says:


    I look forward to your newsletter. Thanks for an easy way to learn.



    It is more important site to learn mathematics.Thank for this excellent site. it is not only for students suitable also for teacher and gives desireable information.

  5. ray says:

    Hi Murray,

    Excellent article in latest subscribed-to Newsletter:

    8 -Radius of Curvature!

    From the formula in the Example 1/Answer 1,
    I understood how you derived
    the radius of the circle as being = 11.047...,
    at x = 1.

    But how did you derive that the (x,y) coords of the CENTER
    of the Circle are: (-9.83, 6.17), at x=1 ?

    (I already subscribe to your newsletter).

  6. Murray says:

    Hi Ray. OK, fair enough question! I have amended the solution for Example 1 and it now explains how I found the center of the circle . See Radius of Curvature.

  7. Ahmed says:

    Dear Murray,

    Your latest newsletter with the topic RADIUS of CURVATURE is most illuminating.

    This is so specially because you have it all as equations displayed as graphs. I wish I were able to do the same for my students.

    This academic year I have 5 students who learnt Applications of Derivatives from me. Two of them were home (here) from Delhi for Summer Vacations. Unfortunately, their institutions have resumed studies starting from
    roughly the middle of this month. We did good work;when it is repeated in class by their real professor, things may sink in more!!

    The 3 who are left are girls who study here at the local Higher Secondary School. We learn in the ancient way: chalk & blackboard + a prescribed TEXT BOOK. No graphs !! No calculators allowed !!! No laptops/ computers. We
    are in the 1970s era.

    The girls studied up to STD 10 in Hindi-medium village schools situated way up in very remote snowfall regions of the Himalayas. Now for Std. 11 & Std 12, for which they have come here to MANALI, the next stage of education are all based on English-language text books in Physics, Chemistry & Math.

    How would your under-grad students fare in their studies if they had to do the last few years to come by the use of text books in French or Chinese or Arabic? They do not speak English. Its only a reading language for them. This , Manali, is a village !! Who speaks English here ?
    Only the elite.

    If you do not know English well, it slows down the process of understanding word-problems in applied calculus and applied Matrices. Their mother-tongues are not Hindi either !!! The home-dialect (not a written anguage) is Lahouli. It is more akin to Tibetan---an indo-mogolian dialect. Some folk from LAHAUL look just like our concept of Gengies Khan. Many have plain Indian features. MIXED !!!

    With best wishes.

    {(I have had a cataract operation last month---on the left eye. So, now my vision is BIBLICAL. The right eye does not know what the left one is focused on !!!)}

    In August, we will operate the other eye.

  8. Murray says:

    Thanks for the interesting story, Ahmed!

    You've raised several interesting things, especially the part about learning math in a language other than the one your use to think.

    Many of the students I've taught have not been native speakers of English (in Australia, Japan and Singapore). It's interesting that while not everyone likes math, it's way better than reading, when you are operating in a strange language!

    And ou also talked about non-calculator and non-computer approaches to teaching math. There are many teachers who would like to go back to such an era! Calculators (and even more so, computers) are often only really useful tools if you know what you are doing on paper first.

  9. Ahmed says:

    As far as up to date technology in learning is concerned, you have everything. But, good math can also be learnt in the rural areas of the 3rd world. All one needs is eager students, a few suitable text books, and dedicated teaching.

    By the way, teaching mathematics does not make one a "creative mathematician"; that's a gift from heaven to one or two in a century.

  10. Murray says:

    No argument from me on those 2 points!

  11. Anesu says:

    Dear Murray,

    Thank you for another edition of The IntMath Newsletter. This time I was impressed by the article on summer brain drain. Forgetting is one of the major challenges here in Namibia yet it very difficult to encourage esspecially junior students to revise and rehearse so that they retain more during exams. I am certainly sure that the graphs of retention and forgetting will assist me in addressing one of the serious cases we have around.

    Thank you for keeping us in the spirit of Math.


  12. Murray says:

    Hi Anesu - glad you found it useful.

    And believe me - forgetting is not only a problem in Namibia!

  13. chinnu says:

    the newsletter is really interesting. the summer brain drain is so true. i have always wondered why that happened. guess i know now.
    could you cover something on the daily uses of parabolas and the innovations concerning the parabola.

  14. ananthamurthy says:

    Dear sir, I have gone through the subject on Radius of curvature which is very interesting. It will make me feel much more interested If you please help with the derivation of the formula on the Radius of curvature. Thanks.

  15. Murray says:

    Hi Ananathamurthy. This Wikipedia article shows the derivation of radius of curvature formula.

  16. augie says:

    thank you for your recomendations which helps me a lot in my mathematics learning..

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